First note that the values x = -8/3x=−83 and t = -1t=−1 cause division by 00, so are excluded values.
Multiply both sides of the equation by (3x + 8)(3x+8) to get:
(3x+8)/(t+1) = 2x-63x+8t+1=2x−6
Multiply both sides of this equation by (t+1)(t+1) to get:
3x+8 = (2x-6)(t+1)3x+8=(2x−6)(t+1)
=(2x-6)t + (2x - 6)=(2x−6)t+(2x−6)
Subtract (2x-6)(2x−6) from both sides to get:
(2x-6)t = (3x+8)-(2x-6)(2x−6)t=(3x+8)−(2x−6)
=3x + 8 - 2x + 6=3x+8−2x+6
=3x - 2x + 8 + 6=3x−2x+8+6
=(3-2)x+14=(3−2)x+14
=x+14=x+14
Divide both ends by (2x-6)(2x−6) to get:
t = (x+14)/(2x-6)t=x+142x−6
=(x-3+17)/(2(x-3))=x−3+172(x−3)
=(x-3)/(2(x-3))+17/(2(x-3))=x−32(x−3)+172(x−3)
=1/2+17/(2(x-3))=12+172(x−3)
=1/2+17/(2x-6)=12+172x−6
So:
color(red)(t = 1/2 + 17/(2x-6))t=12+172x−6
To find xx in terms of tt, first subtract 1/212 from both sides to get:
t - 1/2 = 17/(2x-6)t−12=172x−6
Multiply both sides by 22 to get:
2t-1 = 17/(x-3)2t−1=17x−3
Multiply both sides by (x-3)(x−3) and divide both sides by (2t-1)(2t−1) to get:
x-3 = 17/(2t-1)x−3=172t−1
Add 33 to both sides to get:
color(red)(x = 17/(2t-1)+3)x=172t−1+3
